Chaos Suppression and Stabilization of Generalized Liu Chaotic Control System

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International Open Access Journal
ISSN No: 2456
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Chaos Suppression
Generalized Liu Chaotic Control System
Yeong
Department of Electrical Engineering
ABSTRACT
In this paper, the concept of generalized stabilization
for nonlinear systems is introduced and the
stabilization of the generalized Liu chaotic control
system is explored. Based on the time
approach with differential inequalities, a suitable
control is presented such that the generalized
stabilization for a class of Liu chaotic system can be
achieved. Meanwhile, not only the guaranteed
exponential convergence rate can be arbitrarily pre
specified but also the critical time can be correctly
estimated. Finally, some numerical simulations are
given to demonstrate the feasibility and effectiveness
of the obtained results.
Key Words: Generalized synchronization, Liu chaotic
system, critical time, exponential convergence rate
1. INTRODUCTION
In recent years, chaotic dynamic systems have been
widely investigated by researchers; see, for instance,
[1-12] and the references therein. Very often, chaos in
many dynamic systems is an origin of the generation
of oscillation and an origin of instability.
control system, it is important to design a controller
that has both good transient and steady-state response.
Furthermore, suppressing the occurrence of chaos
plays an important role in the controller design of a
nonlinear system.
In the past decades, various methodologies in control
design of chaotic system have been presented
variable structure control approach,
approach, adaptive control approach, adaptive sliding
mode control approach, back stepping control
approach, and others.
International Journal of Trend in Scientific Research and Development (IJTSRD)
International Open Access Journal | www.ijtsrd.com
ISSN No: 2456 - 6470 | Volume - 3 | Issue – 1 | Nov
www.ijtsrd.com | Volume – 3 | Issue – 1 | Nov-Dec 2018
Chaos Suppression and Stabilization of
Yeong-Jeu Sun1
, Jer-Guang Hsieh2
1
Professor, 2
Chair Professor
Electrical Engineering, I-Shou University, Kaohsiung
he concept of generalized stabilization
for nonlinear systems is introduced and the
stabilization of the generalized Liu chaotic control
system is explored. Based on the time-domain
approach with differential inequalities, a suitable
uch that the generalized
stabilization for a class of Liu chaotic system can be
achieved. Meanwhile, not only the guaranteed
exponential convergence rate can be arbitrarily pre-
specified but also the critical time can be correctly
numerical simulations are
given to demonstrate the feasibility and effectiveness
Generalized synchronization, Liu chaotic
system, critical time, exponential convergence rate
tic dynamic systems have been
; see, for instance,
] and the references therein. Very often, chaos in
an origin of the generation
an origin of instability. For a chaotic
control system, it is important to design a controller
state response.
Furthermore, suppressing the occurrence of chaos
oller design of a
In the past decades, various methodologies in control
presented, such as
time-domain
approach, adaptive control approach, adaptive sliding
stepping control
In this paper, the concept of generalized
for nonlinear dynamic systems is introduced and the
stabilizability of generalized Liu chaotic
system will be investigated
domain approach with differential inequality, a
suitable control will be offered
generalized stabilization can be achieved for a class of
Liu chaotic system. Not only
correctly estimated, but also
exponential convergence rate can be arbitrarily pre
specified. Several numerical simulations will also be
provided to illustrate the use of the main results.
The layout of the rest of this paper is organized as
follows. The problem formulation, mai
controller design procedure are presented
Numerical simulations are given in Section 3
the effectiveness of the developed results.
conclusion remarks are drawn
2. PROBLEM FORMULATION AND MAIN
RESULTS
Nomenclature
n
ℜ the n-dimensional real space
a the modulus of a complex number
T
A the transport of the matrix
x the Euclidean norm of the vector
In this paper, we explore the following generalized
Liu chaotic system:
( ) ( ) ( ) ( ),122111 tutxatxatx ++=&
( ) ( ) ( ) ( ) ( ),2314132 tutxtxatxatx ++=&
( ) ( ) ( ) ( )
( ) ( ) ,0,3
2
18
3726153
≥∀++
++=
ttutxa
txatxatxatx&
Research and Development (IJTSRD)
www.ijtsrd.com
1 | Nov – Dec 2018
Dec 2018 Page: 1112
f
Kaohsiung, Taiwan
generalized stabilizability
dynamic systems is introduced and the
generalized Liu chaotic control
investigated. Based on the time-
domain approach with differential inequality, a
offered such that the
stabilization can be achieved for a class of
system. Not only the critical time can be
, but also the guaranteed
exponential convergence rate can be arbitrarily pre-
. Several numerical simulations will also be
provided to illustrate the use of the main results.
The layout of the rest of this paper is organized as
The problem formulation, main result, and
controller design procedure are presented in Section 2.
given in Section 3 to show
developed results. Finally,
in Section 4.
ROBLEM FORMULATION AND MAIN
dimensional real space
the modulus of a complex number a
the transport of the matrix A
the Euclidean norm of the vector n
x ℜ∈
the following generalized
(1a)
(1b)
(1c)

International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456
( ) ( ) ( )[ ]
[ ] ,
000
302010
321
T
T
xxx
xxx
=
where ( ) ( ) ( ) ( )[ ] 3
321: ℜ∈=
T
txtxtxtx is the state vector,
( ) ( ) ( ) ( )[ ] 3
321: ℜ∈=
T
tutututu is the system
[ ]T
xxx 302010 is the initial value, and ai ,
the parameters of the system. The original
system is a special case of system (1) with
1021 =−= aa , ,403 =a ,14 −=a ,065 == aa
48 =a . It is well known that the system (1) without
any control (i.e., ( ) 0=tu ) displays chaotic behavior for
certain values of the parameters [1]. The
paper is to search a novel control for the system (1)
such that the generalized stability of the
controlled system can be guaranteed. In this
concept of generalized stabilization will be
introduced. Motivated by time-domain approach with
differential inequality, a suitable control
be established. Our goal is to design a control such
that the generalized stabilization of system
achieved.
Let us introduce a definition which will be used in
subsequent main results.
1 There exist two positive numbers k
that ( ) 0, ≥∀⋅≤ −
tektx tb
.
2 There exists a positive number
( ) ctttx ≥∀= ,0 .
Definition 1
The system (1) is said to realize the
stabilization, provided that there exist a suitable
control u such that the conditons (i) and (ii) are
satisfied. In this case, the positive number
the exponential convergence rate and
number ct is called the critical time.
Now we present the main result for the
stabilization of the system (1) via
approach with differential inequalities.
Theorem 1
The system (1) realizes the generalized
under the following control
( ) ( ) ( ) ( ) ( ),12
122111 taxtxatxbatu −
−−+−= α
(2a)
( ) ( ) ( ) ( ) ( )
( ),12
2
2314132
tax
tbxtxtxatxatu
−
−
−−−=
α
(2b)
( ) ( ) ( ) ( ) ( )
( ) ( ),12
3
2
18
3726153
taxtxa
txbatxatxatu
−
−−
+−−−=
α
(2c)
of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456
(1d)
is the state vector,
is the system control,
ℜ∈b, indicate
. The original Liu chaotic
is a special case of system (1) with ( ) ,0=tu
5.27 −=a , and
It is well known that the system (1) without
) displays chaotic behavior for
he aim of this
for the system (1)
of the feedback-
In this paper, the
stabilization will be
domain approach with
suitable control strategy will
a control such
stabilization of system (1) can be
Let us introduce a definition which will be used in
k and b , such
There exists a positive number ct such that
The system (1) is said to realize the generalized
there exist a suitable
such that the conditons (i) and (ii) are
In this case, the positive number b is called
convergence rate and the positive
the generalized
(1) via time-domain
generalized stabilization
where ,0,0 >> ba
12
1
:
−
−+
=
p
qp
α , with
In this case, the pre-
convergence rate and the guaranteed critical time
given by b and
( ) ( )[
( )12
00
ln
2
3
2
2
2
1
b
xxx
b
a
tc
α−−






++
=
respectively
Proof. From (1)-(2), the feedback
can be performed
( ) ,
12
111
−
−−=
α
xabxx&
( ) ,
12
222
−
−−=
α
xabxx&
( ) ,
12
333
−
−−=
α
xabxx&
Let
( )( ) ( ) ( )txtxtxW T
= .
The time derivative of ( )( )txW
feedback-controlled system is given by
2211 22 xxxxW &&& ⋅+⋅=
α2
1
2
2
2
1 2222 axaxbxb −⋅−⋅−⋅−=
( )αα 2
2
2
122 xxaWb +−⋅−=
.0,22 ≥∀⋅−⋅−≤ tWaWb α
It follows that
( ) ( )
( ) .0,12
121 1
≥∀−−≤
−+− −−
ta
bWWW
α
αα αα &
Define
( ) ( )( ) .0,:
1
≥∀=
−
ttxWtQ
α
From (6) and (7), it can be readily obtained that
( ) ( ),1212 ≥∀−−≤−+ tabQQ αα&
It is easy to deduce that
( )
( ) ( )
( ) (
( )
( )[ ]
( ) ( )
.0,12
12
12
12
1212
≥∀−−≤
⋅=
−⋅+⋅
−
−
−−
tea
tQe
dt
d
QbetQe
bt
bt
btbt
α
α
αα
α
α&
It follows that
( )
( )[ ]
( )
( ) ( )012
0
12
QtQe
dttQe
dt
d
bt
t
bt
−⋅=
⋅
−
−
∫
α
α
of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470
Dec 2018 Page: 1113
, with Nqp ∈, and qp >
-specified exponential
guaranteed critical time are
( )]
)
,
0
12
3
b
b
aα






+
−
(3)
feedback-controlled system
(4a)
(4a)
(4a)
(5)
along the trajectories of
is given by
α2
2xa ⋅
(6)
(7)
), it can be readily obtained that
.0
( )t

( ) ( )
( )
( ) .0,1
12
12
0
12
≥∀−
−
=
−−≤
−
−
∫
te
b
a
dtea
bt
t
bt
α
α
α
Consequently, we have
( ) ( ) ( )
.0
,0 12
≥∀
−⋅





+≤ −−
t
b
a
e
b
a
QtQ btα
(8)
Hence, from (6), (7), and (8), we have
(i) If ,0 ctt ≤≤
( )( ) ( ) ( )
( )
;0
11
1222
α
αα
−
−−−






−⋅





+≤
b
a
e
b
a
xtxW bt
(ii) If
( ) 0=tx .
Consquently, we conclude that
(i) If ,0 ctt ≤≤
( ) ( )
( )
;0
221
22 bt
e
b
a
xtx −
−
−
⋅





+≤
α
α
(ii) If ,ctt ≥ ( ) ,0=tx
in view of (5) with above condition (i)
completes the proof. □
3. NUMERICAL SIMULATIONS
Consider the generalized Liu chaotic system
with 1021 =−= aa , ,403 =a ,14 −=a 5 = aa
, ,48 =a and ( ) [ ]T
x 2240 −= . Our objective, in
example, is to design a feedback control such that the
system (1) realize the generalized stabilization
with the guaranteed exponential convergence rate
5.0=b . From (2), with ,3,100 === qpa
8.0=α ,
( ) ( ) ( ) ( ),100105.10 6.0
1211 txtxtxtu −+−= (9a)
( ) ( ) ( ) ( ) ( )
( ),100
5.040
6.0
2
23112
tx
txtxtxtxtu
−
−+−=
(9b)
( ) ( ) ( ) ( ),10042 6.0
3
2
133 txtxtxtu −−= (9c)
Consequently, by Theorem 1, we conclude that
system (1) achieves generalized stabilization
parameters of 1021 =−= aa , ,403 =a 4 =a
5.27 −=a , 48 =a , and feedback control law of (
Furthermore, the exponential convergence rate
guaranteed critical time are given by
047.0=ct .
The typical state trajectories of uncontrolled systems
and controlled systems are depicted in Fig
Figure 2, respectively. From the foregoing simulations
(ii) If ,ctt ≥
with above condition (i). This
Liu chaotic system of (1)
,06 =a 5.27 −=a
Our objective, in this
example, is to design a feedback control such that the
stabilization with
with the guaranteed exponential convergence rate
,2 we deduce
, we conclude that the
achieves generalized stabilization with
,1− ,065 == aa
feedback control law of (9).
exponential convergence rate and the
iven by 5.0=b and
of uncontrolled systems
and controlled systems are depicted in Figure 1 and
, respectively. From the foregoing simulations
results, it is seen that the dynamic system of (
achieves the generalized stabilization under the
control law of (9).
4. CONCLUSION
In this paper, the concept of generalized
for nonlinear systems has been introduced and the
stabilization of generalized Liu chaotic
has been studied. Based on the time
with differential inequalities, a s
been presented such that the generalized
for a class of Liu chaotic system
Besides, not only the guaranteed exponential
convergence rate can be arbitrarily pre
also the critical time can be correctly estimated.
Finally, some numerical simulations have been
offered to show the feasibility and effectiveness of the
obtained results.
ACKNOWLEDGEMENT
The authors thank the Ministry of Science and
Technology of Republic of China for supporting this
work under grants MOST 106
MOST 106-2813-C-214-025-E
E-214-030.
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type-2 fuzzy brain emotional learning control
Dec 2018 Page: 1114
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Figure 1: Typical state trajectories of the system (1)
with ,0=u 1021 =−= aa , 3 =a
5.27 −=a , and
Figure 2: Typical state trajectories of the feedback
controlled system of (
0 5 10 15 20 25
-40
-20
0
20
40
60
80
100
120
t (sec)
x1(t);x2(t);x3(t)
x1: the Blue Curve x2: the Green Curve
0 0.01 0.02 0.03 0.04
-2
-1
0
1
2
3
4
t (sec)
x1(t);x2(t);x3(t)
x1
x2
x3
Dec 2018 Page: 1115
Typical state trajectories of the system (1)
,40 ,14 −=a ,065 == aa
and 48 =a .
Typical state trajectories of the feedback-
controlled system of (1) with (9).
30 35 40 45 50
x2: the Green Curve x3: the Red Curve
0.05 0.06 0.07 0.08

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